A fundamental open question is how these two seemingly very different sets of observables are related.
Intricate examples of mutual compatibility will be encountered throughout this work. But even more than just compatibility should be true: conjecturally, the VOA and Coulomb branch geometry separately provide complete characterizations of the 4d SCFT,This statement concerning the Coulomb branch requires some clarification. Since the early papers on the subject, it has been known that for rank-1 theories where the Coulomb branch stratification data is trivial, the scale-invariant limit of the Coulomb branch is not enough to fully specify the SCFT [1]}. However, it is still possible to enrich this geometric data with other purely Coulomb branch quantities, e.g. the deformation pattern [2]}, to provide (up to discrete gauging) a complete characterization. At higher ranks, the stratification of singular loci becomes considerably more rich and, to the authors' knowledge, no inequivalent SCFTs share the same stratification data, i.e. assignments of low-energy theories on sub-loci of all complex codimensions. in the sense that no two distinct SCFTs map to the same VOA or to the same CB geometry.
It should then in principle be possible to fully reconstruct the CB geometry from the VOA, and vice versa. A tantalizing hint in this direction
is the experimental observation of [3]}: in several cases, one can precisely relate the Schur index
with a certain wall-crossing invariant computed from the massive BPS spectrum on the Coulomb branch.
